Some modest Yahtzee goals

Probability Level 3

If the probability of getting at least one pair of matching dice when rolling five 6-sided dice is $\frac{a}{b}$ , where $a$ and $b$ are positive co-prime integers, then what is $a+b$ ?

Note: Three-of-a-kind, four-of-a-kind, full house, yahtzee, or two pairs would all count as "at least one pair of matching dice." However, a straight ( $\{1,2,3,4,5\}$ or $\{2,3,4,5,6\}$ ) or a small straight would not count.

Check out The anti-Yahtzee first to make finding the solution to this problem trivial.